# Variance of a stationary AR(2) model

I have two questions:

1) When one says an ARMA process is 'stationary,' do they mean strongly stationary or weakly stationary?

2) Is there a quick way to find the variance of a stationary AR(2) model $$y_t = \beta_1 y_{t-1} + \beta_2 y_{t-2} + \epsilon_t?$$ The only way I can think of doing this is by multiplying by $y_t$, $y_{t-1}$ and $y_{t-2}$, taking expectations, and solving the Yule-Walker system with 3 equations and 3 unknowns. The trick for AR(1) models, where one takes expectations of both sides, doesn't slide here because you get a $\mathrm{Cov}(y_{t-1}, y_{t-2})$ term.

• Regarding (1), it must depend on the context. An answer claiming otherwise would be careless, IMHO. – Richard Hardy Jan 19 '17 at 17:39
• Regarding (2), the variance of the AR(2) model will depend on the first and second order autocovariances. As you say, in order to get its value you will have to solve the Yule-Walker equations; by doing so, I think you will get $Var(y_t)=\frac{(1-\beta_2)\sigma^2_\varepsilon}{(1+\beta_2)[(1-\beta_2)^2-\beta_1^2]}$. – javlacalle Jan 19 '17 at 18:10